A Macroeconomic Model of Liquidity, Wholesale Funding and Banking Regulation

Transcription

1 A Macroeconomic Model of Liquidiy, Wholesale Funding and Banking Regulaion Corinne Dubois EPFL Luisa Lamberini EPFL Absrac This paper proposes a model of wholesale funding, liquidiy and liquidiy regulaion in he U.S. banking secor. We develop a dynamic macroeconomic model wih regulaed banks and a shadow banking secor. The banks raise deposis and ge wholesale funding from he shadow bank, bu moral hazard limis heir abiliy o access wholesale funding. Following an increase in bank asse riskiness, moral hazard ighens, banks suffer a wholesale run and are forced o cu loans, which in urn depresses invesmen and oupu. We find ha liquidiy regulaion relaxes he moral hazard faciliaes access o wholesale funding. Neverheless, a fla liquidiy regulaion, as proposed in Basel III, has adverse consequences on he volailiy of oupu and consumpion and amplifies he response o shocks. Liquidiy regulaion ha akes ino accoun he economic cycle is sabilizing and welfare improving. We es empirically he implicaion of our model, and provide evidence ha banks wih a higher liquidiy raio suffer less conracion of wholesale funding and loans during a ime of liquidiy sress. École Polyechnique Fédérale de Lausanne, corinne.dubois@epfl.ch. École Polyechnique Fédérale de Lausanne, luisa.lamberini@epfl.ch. 1

2 1 Inroducion The financial crisis of emphasized he imporance of banks in he macroeconomy and how heir financial weakness quickly ransmis o he real economy. Before he crisis, banks relied on shor-erm wholesale funding, while invesing in illiquid asses wih uncerain valuaion. Banks were hus exposed o liquidiy risk. During he financial crisis banks were rapped in an adverse liquidiy spiral: illiquid asses los value, banks faced higher margins and haircus on heir shor-erm borrowing, making i increasingly difficul for hem o roll i over, and forcing hem o a fire sale of asses, ha furher depressed asse prices. Following he financial crisis, Basel III inroduced liquidiy regulaion for banks aimed a making he banking secor more resilien o liquidiy sress. The impac of bank liquidiy regulaion on aggregae variables such as oupu, credi, invesmen, consumpion and asse prices, is unknown. The goal of his paper is o analyze liquidiy regulaion from a macroeconomic perspecive. This paper proposes a dynamic macroeconomic model wih regulaed banks and a shadow banking secor. The banks raise deposis from households and receive wholesale funding from he shadow bank; hey lend o firms ha are subjec o idiosyncraic risk. There is a moral hazard problem for he banks: because of limied liabiliy, hey have an incenive o inves in riskier bu subopimal firms. The shadow bank recognizes he moral hazard and limis is lending o he bank o ensure ha i invess opimally. Households do no have an incenive o monior he banks because heir deposis are fully insured. An increase in he riskiness of bank asses ighens moral hazard and causes a run on wholesale funding. The wholesale run leads o a credi crunch and a fall in invesmen and oupu. In his seing we inroduce a liquidiy regulaion ha forces he bank o hold a fracion of safe and highly liquid asses and sudy he effecs on he macroeconomy. The liquidiy regulaion changes he seady sae of he model: banks hold safe asses, so heir porfolio of asses becomes safer. This relaxes heir moral hazard and enables hem o leverage up and lend more o firms. Thus, liquidiy crowds in loans, and he regulaed seady sae feaures more credi, higher oupu, invesmen and consumpion. In a dynamic seing, we sudy fla and counercyclical liquidiy regulaions. Fla regulaion requires banks o hold a consan fracion of liquid asses agains 2

3 deposis and wholesale funding, and is in line wih Basel III liquidiy coverage raio. Counercyclical regulaion goes beyond Basel III, and requires banks o hold on o more liquid asses during a recession. We find ha fla liquidiy regulaion has unaended consequences on he volailiy and persisency of macroeconomic variables following a shock, and is welfare implicaion is ambiguous. On he oher hand, counercyclical liquidiy regulaion, has a sabilizing effec on he economy and is unambiguously welfare improving. This resul may seem counerinuiive a firs; bu inuiively, holding on o more liquid asses makes banks safer and reduces wholesale bank runs. Our model predics ha banks ha hold a higher raio of liquid asses will suffer a smaller conracion of wholesale funding during a ime of liquidiy sress. In he second par of his paper, we ake he implicaion of our model o he daa. We use he TED spread as a proxy for marke liquidiy sress and inerac i wih he liquidiy raio of banks. We hen run a regression of he growh of wholesale funding on he liquidiy raio and he ineracion erm of liquidiy and TED spread. We find ha he coefficien on he ineracion erm beween he liquidiy raio and he TED spread is posiive and significan. This implies ha when he TED spread is high, banks wih a higher liquidiy raio suffer less of a run on heir wholesale funding. 2 Lieraure review The lieraure saring wih Diamond and Dybvig 1983) emphasizes he role of banks as liquidiy providers. These auhors show ha bank liquidiy provision improves he economic oucome, bu can also be subjec o harmful bank runs, hence he crucial role of deposi insurance. Diamond and Rajan 2, 21) furher show ha bank fragiliy resuling from acceping demand deposi is an essenial feaure of he bank. Demand deposis are a disciplining mechanism for bankers and makes i possible for hem o lend more, which is beneficial o he economy. Angeloni and Faia 213) inroduce banks à la Diamond and Rajan 21) in an dynamic macroeconomic model. More recenly, Gerler and Kiyoaki 215) develop a macroeconomic model wih banks and sunspo bank-run equilibria. In all hese papers, he main risk for bank is a run by deposiors. However, he exisence of deposi insurance makes deposis a relaively safe source of funding 3

4 for he bank. In our model, deposis are safe and he risk for he bank comes from wholesale funding. A second group of conribuions examines he role of financial fricions in he ransmission of shocks o he macroeconomy. An early conribuion on financial fricions is Bernanke e al. 1999). The financial fricion is asymmeric informaion beween borrowers and lenders and a cosly sae verificaion. The fricion gives rise o a counercyclical exernal finance premium, leading o an amplificaion of he business cycle. In his model banks are a veil. Gerler and Kiyoaki 21) and Gerler e al. 212) develop a dynamic macroeconomic model wih financial inermediaion. Banks are subjec o moral hazard due o heir abiliy o diver a fracion of asses. The fricion gives rise o an endogenous balance shee consrain ha amplifies he effecs of shocks. He and Krishnamurhy 213) and Brunnermeier and Sannikov 214) propose a coninuous-ime nonlinear macroeconomic model wih a financial secor. Brunnermeier and Sannikov 214) model generaes a srong amplificaion o large shocks ha hi he ne worh of he financial inermediaries, forcing hem o fire-sell heir capial. In He and Krishnamurhy 213) financial expers face an occasionally binding equiy capial consrain. The equiy consrain relaes how much equiy a bank can raise o is repuaion, which is a funcion of pas performance. The equiy consrain only binds during sysemic crises, so he model feaures high amplificaion of shocks during crises. Our approach is differen because we inroduce regulaion on financial inermediaries. Our heoreical work is relaed o Adrian and Shin 214). The auhors provide empirical evidence on invesmen banks and show ha he leverage raio is procyclical, whereas Value-a-Risk over equiy remains relaively sable over he business cycle. Value-a-Risk VaR) is a quanile measure on he loss disribuion defined as he smalles hreshold loss L such ha he probabiliy ha he realized loss urns ou o be larger han L is a mos α. Their evidence suppors he idea ha banks acively manage heir leverage raios in order o mainain heir equiy equal o heir VaR. If a bank manages is risk by keeping is VaR smaller han is equiy, hen i ensures ha i can absorb losses up o he level of he VaR and i will be solven wih a probabiliy of a leas 1 α. The auhors propose a heoreical model wih banks ha borrow from a wholesale credior. Banks have he choice beween invesing in a producive projec and an inefficien projec wih lower expeced payoff bu higher volailiy. Due o limied liabiliy, he bank has 4

5 an incenive o inves in he inefficien projec because i has higher upside risk. The incenive o inves inefficienly increases wih he leverage of he bank, so he conracing framework resuls in an endogenous leverage raio for he bank. Nuño and Thomas 213) implemen he conrac proposed by Adrian and Shin 214) in a dynamic macroeconomic model. Their model explains bank leverage cycles as he resul of risk shocks, namely of exogenous changes in he volailiy of idiosyncraic risk. Inuiively, higher volailiy makes i more aracive for banks o inves inefficienly, so he wholesale credior reduces lending o induce banks o inves efficienly. Following a risk shock, banks are forced o deleverage; as a resul he price of bank asses falls, inducing a decline in invesmen and oupu. Our work builds on Adrian and Shin 214) and Nuño and Thomas 213), which we exend in several imporan ways. Firs, our banks choose heir liabiliy srucure, namely hey choose beween deposis and wholesale funding, and hey can defaul on deposiors. Second, our banks are regulaed and subjec o deposi insurance by he governmen. Several empirical papers emphasize he imporance of wholesale funding and liquidiy in causing or amplifying he recen crisis. Raddaz 21) performs an even sudy on 662 banks commercial and invesmen) from 44 counries, where he liquidiy crunch even is he failure of Lehman Brohers on Sepember 15 h 28. Raddaz 21) finds ha banks ha relied more heavily on wholesale funding before he even experienced a significanly larger decline in sock reurns following he liquidiy crunch. Dagher and Kazimov 212) provide empirical evidence on he effec of wholesale funding on he supply of lending during he crisis. They use loan-level daa from 25 o 28 and es wheher banks ha are more relian on wholesale funding have a higher rae of rejecion of loan applicaions. Afer conrolling for bank, borrower, and regional characerisics resuls show ha banks ha more relian on wholesale funding experienced a significanly larger conracion in he volume of credi in 28. Ivashina and Scharfsein 21) and Corne e al. 211) analyze he impac of bank vulnerabiliy o liquidiy risk on lending during he financial crisis. Liquidiy risk is defined as exposure o wholesale funding and drawdown from credi lines. Ivashina and Scharfsein 21) uses he percenage of credi lines ha are co-syndicaed wih Lehman Brohers o approximae for unexpeced drawdowns on credi lines and find ha banks wih higher unexpeced drawdowns on credi lines and higher wholesale funding cu 5

6 credi more during he financial crisis. Corne e al. 211) use he TED spread as an indicaor of liquidiy sress and inerac i wih deposi, equiy and unused commimen raios. Their regression suggess ha banks wih higher liquidiy risk experience sronger negaive effec on lending when he TED spread is high. Our empirical analysis builds upon Corne e al. 211). We analyze he impac of holding liquid asses on he growh of wholesale funding and loans using he TED spread o proxy for liquidiy sress. 3 Basel III Liquidiy regulaion The aim of Basel III is o srenghen capial and liquidiy rules o make he banking secor more resilien o shocks. Basel III requires higher capial raios compared o previous regulaions and i inroduces wo liquidiy requiremens: he liquidiy coverage raio and he ne sable funding raio. The liquidiy coverage raio LCR) is he sock of high qualiy liquid asses HQLA) over he oal ne cash ouflow over he nex 3 days. Basel III requires his raio o be a leas 1%. The goal is o ensure ha he bank has enough liquid asses o wihsand a 3-days liquidiy sress scenario. In order o qualify as HQLA, asses should be liquid in markes during a ime of sress and, in mos cases, be eligible for use in cenral bank operaions. Cerain ypes of asses wihin HQLA are subjec o a range of haircus. Expeced cash ouflows are calculaed by muliplying he ousanding balances of various ypes of liabiliies and off-balance shee commimens by he raes a which hey are expeced o run off or be drawn down under a sress scenario. Toal ne cash flow over he nex 3 days is defined as oal expeced cash ouflows, minus oal expeced cash inflow. There is a cap on oal inflows 75%) o avoid ha banks rely only on anicipaed inflows o mee heir liquidiy requiremens. Toal ne cash flow over he nex 3 days is hen calculaed as follows: Toal ne cash flow = ouflows mininflows, 75% of ouflows). 1) In he Unied Saes, he Federal Reserve Board issued a final version of he LCR in Sepember 214. The LCR is o be phased in beween 215 and 217. The ne sable funding raio NSFR) is defined as: Available amoun of sable funding Required amoun of sable funding 6 1%. 2)

7 The goal of he ne sable funding raio is o ensure ha a cerain fracion of long-erm asses is funded wih sable liabiliies and avoid over-reliance on shorerm funding when marke is buoyan. The available sable funding is he porion of equiy and liabiliy expeced o be reliable over a one-year ime horizon under condiions of exended sress. The required sable funding depends on he liquidiy characerisics of he asses. I is calculaed as he sum of he value of he asses, muliplied by a specific required sable funding facor assigned o each paricular asse ype. The NSFR is no currenly implemened in he Unied Saes. 3.1 Definiions Our definiion of liquid asses closely follows Basel III. Liquid asses as asses ha are liquid in markes during a ime of sress and can be easily and immediaely convered ino cash a lile or no loss of value. We include cash and excess federal funds reserves as well as reasuries. We also include securiies issued or guaraneed by governmen sponsored agencies 1 henceforh agency securiies). Agency securiies are however subjec o a haircu of 15% and cap of 4%. This means he share of agency securiies o oal liquid asse should no exceed 4%. We hen adjus liquid asses by subracing a fracion of he securiies sold under agreemen o repurchase repo). Liquid asses should be unencumbered, and herefore no par of a repo agreemen. We do no have precise informaion on wha securiies are pledged o repos, so we subrac a fracion of repos equal o he fracion of liquid securiies over all securiies. See Daa appendix for deailed calculaions. Wholesale funding is funding for he banks oher han reail deposis and equiy. We include shor-erm commercial papers issued by he banks, brokered or foreign deposis, repos, inerbank loans, or any oher ype of borrowing. We also consider a measure ha capures liquidiy mismach on bank balance shee. Following Choi and Zhou 214), we build a liquidiy sress raio LSR) for he banks. The LSR is a raio of liquidiy-adjused liabiliies and off-balanceshee iems, and liquidiy-adjused asses. Liquidiy-adjused asses is a weighed average of bank asses where more liquid asses have higher weighs. Liquidiy- 1 Governmen sponsored agencies are he Federal Naional Morgage Associaion Fannie Mae), he Federal Home Loan Morgage Corporaion Freddie Mac) and he Governmen Naional Morgage Associaion Ginnie Mae) 7

8 weighed liabiliies and off-balance-shee iems are also weighed averages where he weighs are smaller for more sable sources of funding. A higher value of he LSR indicaes ha a bank holds relaively more illiquid asses and relaively less sable funding and i is herefore exposed o more liquidiy mismach. The liquidiy, wholesale funding and deposi raios are calculaed by dividing he relevan measure by oal asses. Leverage raio is oal asses divided by equiy. We give a deailed descripion of our ime series and calculaion in he daa appendix. 4 Sylized facs Table 1 shows summary saisics for he leverage, wholesale funding and deposi raios as well as he LSR. We use bank holding company BHC) balance shee daa from he Federal Reserve Y-9C repor FRY9C) from 1994Q1 o 212Q4. We use a sample ha excludes 213 and 214 because we wan o look a daa ha is no affeced by he new liquidiy regulaion. The LCR was finalized in he U.S. in 214, so we argue ha bank behaviour in 213 and 214 may reflec anicipaion of regulaion. The average of he balance shee raios is calculaed by firs aking he average for every quarer, and hen averaging over all periods. The sandard deviaion is calculaed in he same way, by firs aking he average of he raios for each quarer and calculaing is sandard deviaion across ime. To make he second momens comparable wih hose in he model, all variables are in logs. The real GDP and populaion daa comes from he Federal Reserve Bank of S Louis Economic Daa FRED); For real GDP, we calculae a per capia value, ake he log and hp-filer i. Then we calculae he correlaion beween he log of balance shee raios and hp filered GDP; he correlaion are repored in he las column of able 1. Wholesale funding and deposis: Wholesale funding raio is on average 12%. Figure 1 displays he evoluion of wholesale funding, equiy and deposi raios. Noe ha in all he figures, he verical red line corresponds o he hird quarer of 28, when Lehman Brohers collapsed and i is he peak of he financial crisis. Since he mid-9s and unil he financial crisis, banks reliance on deposis has fallen while reliance on wholesale funding has 8

9 increased. The advanage of wholesale funding is ha i is flexible, which allows banks o increase lending. The main drawback is ha i exposes he bank o wholesale runs. Because of deposi insurance, deposiors have lile incenive o monior he banks. Wholesale lenders, on he oher hand, are much more concerned abou banks financial healh because heir deb is no insured. Wholesale lenders can rapidly curail funding if financial condiions change. This sudden reversal can be seen clearly in figure 1: he wholesale funding raio falls dramaically afer he collapse of Lehman Brohers. By he end of 214, he wholesale funding raio had no reurned o is precrisis level. The picure does no enable us o deermine wheher he fall in wholesale funding was driven by a fall in demand for wholesale funding by he banks or by a fall in supply. Goron and Merick 21) sugges ha he fall in wholesale funding was due o a run by wholesale invesors, hence supporing he view of a supply-side reducion. The deposi and equiy raios have increased significanly afer he crisis. Banks urned o safer sources of funding afer he crisis, wheher by choice or hrough he effec of banking regulaion. Table 1 shows ha deposis are considerably less volaile han wholesale funding; his evidence poins owards sickiness of deposis, which is a feaure ha will help us calibrae our model. Moreover, we noe he wholesale funding raio is procyclical whereas he deposis raio is counercylical. Liquidiy: Liquidiy raio is abou 1% on average. Figure 2 shows he evoluion of he average liquidiy raio. The average liquidiy raio decreased persisenly since he mid-9s and hi is lowes value before he crisis. Following he financial crisis banks sared hoarding liquidiy and he liquidiy raio increased sharply. Given he very low liquidiy held by he banks before he crisis, we argue liquidiy regulaion would have been binding had i been in place a ha ime. LSR: Figure 3 displays he evoluion of he LSR, which measures liquidiy mismach on bank balance shee. For consisency reasons, he LSR is only calculaed since 21Q1, as here has been a change in reporing a ha dae. The LSR peaks righ before he financial crisis and hen falls dramaically. The figure clearly shows he build-up of liquidiy risk on bank balance shee 9

10 during he years leading o he financial crisis and he sudden reversal. The LSR is srongly procyclical, as shown in able 1. This suggess ha banks build up liquidiy mismaches during good imes, which can be a major source of financial insabiliy during recessions, when banks suddenly need o reduce such mismach. Leverage: leverage is on average 1.8. Consisenly wih Adrian and Shin 214) and Nuño and Thomas 213), we find he leverage is procyclical see able 1). The firs and second momens of he daa will laer be used o calibrae our model. We will also analyze wheher our model is able o replicae he behaviour of he balance shee raios. Table 1: Summary saisics balance shee raios Mean Sdv Cyclicaliy Deposi raio ** Wholesale raio ** Liquidiy raio ** LSR ** Leverage * * p <.1, ** p <.5, *** p <.1 5 Model Our model builds upon Nuño and Thomas 213). We exend heir model by allowing banks o raise deposis direcly from households and o choose heir liabiliy srucure opimally. We also inroduce deposi insurance and liquidiy regulaion on he banks. 1

11 Figure 1: Wholesale funding evoluion Wholesale, deposi and equiy raios q1 2q1 25q1 21q1 215q equiy deposi righ axis) wholesale Figure 2: Liquidiy raio evoluion Average liquid asses raio 1995q1 2q1 25q1 21q1 215q1 5.1 Households There is a represenaive household ha maximizes uiliy. The household chooses how much o consume C ), supplies L hours of a a wage w. Household can save using insured bank deposi D ), reasury bills T B h ) and shadow bank equiy 11

12 Figure 3: Liquidiy sress raio Average Liquidiy Sress Raio q1 25q1 21q1 215q1 M ). max C,D,M,L,T B = [ C 1 γ β 1 γ η L1+ϕ s.. C + D 1 + χ d 2 ] 1 + ϕ D D ) ) 2 + M + T B h = L w + R H D, 1D 1 + R M, M 1 + R T B, 1 T B h 1 + Π T, 3) where γ is he ineremporal elasiciy of subsiuion, η is he disuiliy from labor and ϕ is he inverse of he Frisch elasiciy of labor supply. Π are ne ransfers from he banks and he capial producers and T are lump sum axes. The rae of reurn on deposis RD, 1 H ) and reasury bills RH T B, 1 ) are perfecly safe and predeermined, which is why we index hem by 1. Shadow bank equiy is risky, he reurn on he shadow bank equiy R M, ) is sae coningen, so i is indexed by. The household faces quadraic adjusmen coss o deposis; hese coss capure he fac ha deposis are sicky, as documened by Dinger and Craig 213) and Huang and Ranovski 211). The idea is ha households hold deposis D) in seady sae and changing he allocaion implies coss ha can be inerpreed as coss of opening a new accoun, or swiching o anoher bank ec. χ d measures he exen of deposi adjusmen coss. 12

13 5.2 Firms Firms are perfecly compeiive and produce he final good Y j and labor L j, subjec o echnological level Z : using capial K j Y j = Z Ω ω j K j ) α L j ) 1 α 4) Firms are segmened across a coninuum of islands indexed by j [, 1] and are subjec o idiosyncraic capial qualiy shocks ω j ha change heir effecive capial o ω j K j. There is also an aggregae capial qualiy shock Ω. Firms are risk-neural and maximize heir operaing profi Π firm,j. Their maximizaion problem is given by: max Π firm,j = Z Ω ω j K j ) α L j L ) 1 α w L j. 5) A ime 1, firms purchase capial K j a price Q 1 from capial producer. They finance heir purchase of capial using loans from banks. They can only borrow from a bank ha is siuaed on he same island; hence, heir balance shee consrain is given by K j +1 = A j. Afer producion akes place, firms sell back depreciaed capial o capial producer. Loans from banks are perfecly saeconingen, so a ime firms use heir profis as well as he proceeds from he sale of depreciaed capial o repay he banks. On each island here are wo ypes of firms, sandard and subsandard. The wo ypes only differ in he disribuion of he idiosyncraic shock: he sandard and subsandard disribuions a ime are respecively F 1 ω) and F 1 ω). Following Nuño and Thomas 213) we assume ha he disribuion of he idiosyncraic shocks are known one period in advance, so hey are indexed by 1.The subsandard disribuion has a lower mean bu higher variance han he sandard one. Subsandard firms never operae in equilibrium bu creae a moral hazard problem for he banks. The disribuions for he island-specific capial qualiy shocks of he sandard and he subsandard firms are given by: ) logω) iid σ 2 N 2, σ, log ω) iid υσ 2 N ϑ, ) υσ. 2 The parameer ϑ > capures he difference in he mean and he parameer υ > capures he difference in he variance beween subsandard and sandard disribuions. 13 6)

14 5.3 Capial producer There is a represenaive capial producer. The capial producer buys he final good in amoun I a a price of one, and ransforms i ino new capial subjec o adjusmen coss S I I 1 ). The new capial is hen sold a a price Q. The capial producer chooses invesmen opimally o maximize is expeced profis. The maximizaion problem of he capial producer is: max E Λ,+1 Q 1 S I ) ))I I, 7) I I 1 = where we assume quadraic adjusmen coss of he form: S I ) = χ ) 2 I 1. 8) I 1 2 I 1 The law of moion of capial is herefore: K +1 = 1 S I ) ) I + 1 δ)ω K. 9) I 1 Profis of he capial producer Π cap ) are ransferred o he households in a lumpsum fashion. 5.4 Banks Banks are also segmened across islands. They raise deposis from households D j ), ge wholesale funding B j ) from he shadow banks, and accumulae ne worh N j ). On heir asse side, banks make loans o firms ha are locaed on he same island Q A j ), and hey hold a risk-free bond issued by he governmen T B j ). Wholesale funding is modeled as uncollaeralized shor-erm borrowing: he bank borrows an amoun B j a ime and promises o he shadow bank a paymen of B j he following period. If he bank has enough asses, i repays he full amoun. If he bank defauls, deposiors are paid off firs and he remaining asses are seized and liquidaed by he shadow bank. The balance shee of he bank is: The cash inflow a ime of banks is given by R A ω j Q 1 A j 1 + R T B, 1 T B j 1, 1) 14

15 Table 2: Balance Shee of bank j Asses Q A j T B j Liabiliies D j B j N j where R A is defined as: R A Rk + 1 δ)q Q 1 Ω. 11) Therefore a bank will defaul if: R A ω j Q 1 A j 1 + R T B, 1 T B j 1 < R B D, 1D j 1 + B j 1. 12) Thus, we can calculae he defaul hreshold ω j : ω j = RB D, 1 Dj 1 R T B, 1 T B j 1 + B j 1 R A Q 1 A j. 13) 1 Banks on islands where he realizaion of ω is above ω repay he full amoun of wholesale funding and deposis whereas banks on islands where he realizaion of he idiosyncraic shock is below ω defaul. We now define a second hreshold ω j. This is he hreshold under which he realizaion of ω is so low ha he bank does no have enough asses o pay back deposiors. In his case, he shadow bank ges nohing and he governmen has o make deposi insurance paymen o deposiors o compensae for he shorfall. This hreshold is given by: ω j = RB D, 1 Dj 1 R T B, 1 T B j 1 R A Q 1 A j. 14) The moral hazard problem of banks As explained in secion 5.2, on each island here are wo ypes of firms: sandard and he subsandard. The bank can choose o inves in eiher one or he oher. The subsandard firm has lower expeced reurn bu higher volailiy compared o he sandard one. When choosing where o inves, he bank faces a he 15

16 rade-off: i prefers an invesmen wih higher expeced reurn, bu i also prefers an invesmen wih higher volailiy. The bank has limied liabiliy: for very low realizaions of he idiosyncraic shock, he bank defauls and he losses are aken by he crediors. However, for large posiive realizaions of he shock, i is he bank ha akes he profis. Thus, he bank prefers volaile invesmens o ake advanage of he upside risk. Noe ha he bank holds a shor posiion in a defaulable deb. As explained in Adrian and Shin 214), defaulable deb wih face value B can be replicaed by a porfolio wih a safe deb B minus a pu opion wih srike price B. Thus, by being shor on he defaulable deb, i is as if he bank owed a non-defaulable amoun B ogeher wih a long posiion in a pu opion value wih srike price B. To see i more clearly, consider he bank expeced profi: ω j +1 R+1Q A A j ω + R T B, T B j B ) j RD,D B j df ω). 15) Replacing B j using he definiion of ω j, we ge: = R A +1Q A j ω j +1 Rearranging expression 15 can be wrien as: ω ω j +1)dF ω). 16) = R A +1Q A j ω j Eω) ω j ω j +1 ω)df ω) }{{} π ω j +1 ) ). 17) π ω j +1) is he value of a pu opion wih srike price ω j +1. Under our disribuional assumpions, we have ha π ω j +1) > π ω j +1), bu Ẽω) < Eω). So when choosing beween sandard and subsandard, he bank rades off higher expeced reurn and lower pu opion value. The pu opion value increases wih risk. Hence, an increase in he riskiness of bank asses makes he moral hazard problem more severe: banks have a higher incenive o inves subopimally. The pu opion value also increases wih leverage. Oher hings being equal, higher leverage implies a higher defaul hreshold ω j +1. As a resul, he srike price of he pu opion is higher, so is value is also higher. I hen follows ha in order o keep he bank from invesing subopimally, is leverage 16

17 canno be oo high. I is exacly by conrolling bank leverage via wholesale funding ha he shadow bank can force he banks o inves in he sandard firms. The moral hazard problem of he banks is a key mechanism in his model: an increase in he riskiness of bank asses makes he moral hazard more severe. The shadow bank recognizes his and responds by reducing wholesale funding Deposi insurance In he Unied Saes he Federal Deposi Insurance Corporaion FDIC) insures deposis up o $25 per person, per insured bank. Banks pay quarerly insurance premiums o he FDIC. The premium depends on he risk aken by banks and how well hey are capialized. Riskier banks pay a higher rae of deposi insurance. The FDIC keeps he fees colleced from he banks in a fund called he Deposi Insurance Fund DIF). The FDIC has a arge for he fund; he fund has o cover a cerain percenage of all insured deposis in he U.S. In he years prior o he financial crisis, he arge for he DIF was 1.25% of insured deposis. The FDIC can rebae fees when he fund is above arge, and can decide o increase fees when he fund is oo low. Hence, deposi insurance fees are procyclical, hey increase when he DIF is drawn down, when here are many bank defauls 2. In our model, deposi insurance is provided by he governmen. The governmen collec deposi insurance fees from he banks and covers losses o households in case banks defaul on heir deposis. From he households perspecive, he deposis are effecively safe. However, banks have o pay a deposi insurance fee on heir deposis. We assume a deposi insurance fee DI ) proporional o he probabiliy of defaul on deposis, o capure he fac ha FDIC fees increase wih risk. Moreover, we accoun for he procyclicaliy of he deposi insurance fee by adding a erm ha capures how far he defaul hreshold is from seady sae. DI = 1 + ι + E ) ω +1 ) ω ss E F ω +1 )), 18) ω ss where ι is a posiive consan and ω ss is he seady-sae value of ω. Banks pay an ineres rae R B D, on heir deposis ha includes he deposi insurance fee. The 2 For a hisory of deposi insurance in he U.S., see Pennacchi 29) 17

18 equilibrium condiion for deposi raes is: R B D, = R H D,1 + DI ). 19) Liquidiy regulaion on banks We inroduce a liquidiy regulaion on banks in he spiri of he liquidiy coverage raio of Basel III. The liquidiy coverage raio specifies ha he bank mus hold sufficien liquid asses o wihsand a 3-day sress scenario. In he model, he highly liquid asse is he governmen bond. The sress scenario is a deposi wihdrawal of ξ, and a run on wholesale funding of ξ 1,. The coninuing bank j mus herefore be able o wihsand such a sress scenario a all imes by holding sufficien liquid asses. Hence he Liquidiy Coverage Raio consrain is: T B j ξ, D j + ξ 1, B j. 2) We firs consider an LCR-ype fla regulaion where ξ and ξ 1 are consan; hen we move beyond he LCR and propose a counercylical liquidiy regulaion where he coefficien ξ and ξ 1 vary wih he business cycle as follows: ξ, = ξ χ y, Y Ȳ ), ξ 1, = ξ 1 χ y,1 Y Ȳ ), 21) where Ȳ is seady-sae oupu and χ y,, χ y,1 are posiive consans. The idea of he counercyclical regulaion is ha during economic downurns, he sress scenario wihdrawals of deposis and wholesale funding becomes bigger for all banks. Thus, banks should be forced o hold on o more safe asses The maximizaion problem of banks To ensure ha banks do no accumulae ne worh indefiniely and cease o rely on ouside funding, we assume ha a fracion 1 ɛ) of non-defauling banks close down every period for exogenous reasons, in which case hey ransfer heir accumulaed ne worh o households. This ransfer from he banking secor o he household can be inerpreed as dividend paymen. The exiing banks are hen replaced by new banks ha receive a ransfer from households. 18

19 Banks maximize heir value. The maximizaion problem of a coninuing bank j can hen be wrien as a Bellman equaion: V N j ) = max E Λ,+1 A j,bj, B j,dj,t Bj ) ɛv+1 N+1) j + 1 ɛ)n j +1 df ω) 22) subjec o five consrains. The firs is he balance shee consrain: ω j +1 The second is he flow of funds consrain: Q A j + T B j = N j + B j + D j. 23) N j +1 = R A +1ω j +1Q A j + R T B, T B j R B D,D j B j, 24) which specifies he evoluion of ne worh of non-defauling bank j. The hird is he liquidiy coverage raio consrain: T B j ξ, D j + ξ 1, B j. 25) The fourh is he incenive compaibiliy consrain ICC): expeced value from invesing in sandard firm { }}{ ) E Λ,+1 ɛv+1 N+1) j + 1 ɛ)n j ω j +1 df ω) +1 ) E Λ,+1 ɛv+1 N+1) j + 1 ɛ)n j ω j +1 d F ω) +1 }{{} expeced value from invesing in subsandard. firm 26) The incenive compaibiliy consrain saes ha he value of he bank when invesing in he sandard firms disribuion F ω)) mus exceed he value of he bank when invesing in he subsandard segmen disribuion F ω)). The fifh is he paricipaion consrain of he shadow bank PC): defaul no defaul {}}{ {}}{ ω j E Λ,+1 B j 1 F ω +1)) j + R+1Q A A j +1 ω +1 df ω) ω j +1 F ω +1) j F ω +1) ) ) j RD,D B j R T B T B j ) E Λ,+1 R T B, B j. }{{} defaul 27) 19

20 The shadow bank has an ouside opion: inves in reasury bills insead of lending o he bank. The paricipaion consrain insures ha he shadow bank is willing o lend o he bank. The expeced reurn from lending o he bank mus exceed he rae on reasury bills. The reurn of he shadow bank will be explained in he nex secion. 5.5 The Shadow Bank The shadow bank issues equiy o households M ) and lends o banks in he form of shor-erm deb B ). Shadow bank equiy is risky because of he defaul risk of he banks. The shadow bank is 1% equiy funded; is payoff from lending o a bank j is: if ω ω, shadow bank receives full paymen B; if ω < ω < ω, deposiors are paid in full and shadow bank seizes he remaining asses of he defauling bank; if ω ω, shadow bank ges nohing. We can wrie he payoff of he shadow bank form lending o bank j as: min B j, max, R+1ω A +1Q j A j + R T B, T B j R D, D j )). 28) Aggregaing across all banks, he gross reurn o he shadow bank is: defaul no defaul { {}}{ }}{ ω+1 R M,+1 M = B j 1 F ω +1 )) + R+1Q A A j ω +1 df ω) ω +1 ) F ω +1 ) F ω +1 )) RD,D B j R T B T B j ). }{{} defaul 29) Noe ha bank liquidiy T B j has a posiive impac on he reurn o he shadow bank. Oher hings being equal, he shadow bank is more relaxed when lending o a regulaed bank because in case of defaul i can seize he safe asses of he bank, which reduces is loss. Deposis on he oher hand have a negaive impac on he reurn of he shadow bank. The reason is ha deposiors are a paid firs in case of defaul, so here are less asses available for he shadow bank o cover is losses. 2

21 5.6 The Governmen The governmen issues he safe asse T B ), provides deposi insurance and raises lump-sum axes T. Is budge consrain is as follows: T B supp + T + Ins fee = R T B, 1 T B supp 1 + Ins pay, 3) where T B supp is he oal supply of reasury bills, which we assume o be fixed. Ins fee are insurance deposi fees colleced from banks and Ins pay are deposi insurance payou o households. We have: Ins fee = RD,D H 1 + ι + E ) ω +1 ) ω ss E F ω +1 ), ω ss Ins pay = R H D, 1D 1 F 1 ω ) and reasury bills are eiher held by he households or by he banks: 5.7 Soluion and aggregaion 31) T B supp = T B h + T B. 32) A soluion o he model is an equilibrium where banks, households, firms and capial producers are opimizing and all markes clear. Following Nuño and Thomas 213), we guess and verify he exisence of a soluion where bank balance shee raios and defaul hresholds are equalized across all islands. Banks in differen islands are differen in erms of size, bu hey all choose he same leverage, deposi, wholesale funding and safe asse raios. We can hen aggregae he banking secor. Aggregaing he flow of funds consrain across all coninuing banks we find ha he evoluion of aggregae ne worh of coninuing banks is given by: N con = ɛr A Q 1 A 1 ω ω ω )df 1 ω). 33) Banks can exi eiher because hey defaul or because hey are hi by he exogenous probabiliy of closing down 1 ɛ). Every period, new banks ener o replace hose ha have exied. Each new bank receives a ransfer of τq A 1 + T B 1 ) from households. The ransfer corresponds o a fracion τ of oal asses in he banking secor a he beginning of he period. We assume ha he new 21

22 banks sar wih he same balance shee raios as he coninuing banks. The oal ne worh of new banks is herefore: N new = [1 ɛ 1 F 1 ω ))] τq A 1 + T B 1 ). 34) Ne ransfers from banks o households Π banks ) are he ne worh ransfers from exiing banks minus he ransfers from households o new banks: Π banks = 1 ɛ) N con N new. 35) ɛ Toal ransfers o households are given by he profi from he capial producers and he ransfer from he banks: Π = Π banks + Π cap. The firs order condiions of all he agens are given in Appendix A. The model can hen be reduced o a se of 24 dynamic equaions ha fully deermine i and are given in Appendix B. 6 Quaniaive analysis 6.1 Calibraion The sandard RBC parameers α, β, γ, δ, χ, ϕ, η) are se in line wih he macro lieraure. Seady-sae level of echnology z is calculaed o normalize seadysae oupu o 1. The share of asse ransfer ino new banks τ is se o arge an invesmen-oupu raio of 2%. Our model economy is calibraed o ge seadysae values of he bank balance shee raios in line wih he daa. The seadysae idiosyncraic volailiy σ is calibraed o arge a leverage raio of 1, and he variance of he subsandard echnology υ is se o arge a wholesale funding raio of 1%. The financial raios in he daa and in he model are in able 3. The Liquidiy Sress Raio does no closely mach he daa, probably because our model bank balance shee is simple and does no encompass he rich se of asses, liabiliies and off balance shee iems ha ener he calculaion of he empirical LSR. The scaling facor of shocks ς and he parameer for deposi sickiness χ d and are se o mach he volailiies of oupu and deposi raio respecively. The volailiies of he macroeconomic and financial variables in he model are repored in able 4 and are close o heir empirical counerpars. The oher parameers θ, ϑ, ρ z, σ z, ρ σ, σ σ, ρ κ, σ κ ) are aken direcly from Nuño and Thomas 213). The 22

23 full calibraion is given in able 5. Basel III specifies ha under a sress scenario, he run-off rae on deposis is 5%, so his is he number we use for calibraing he liquidiy regulaion in he model. In he regulaed version of he model, banks are required o hold 5% of liquid asses agains heir deposis and wholesale funding. In he version of he model wih counercyclical regulaion, banks are required o hold an addiional.5% of liquid asses for every percenage poin of GDP below seady sae. All he regulaory parameers are se o zero in he unregulaed model. The oal supply of reasury bills is se a 2, i.e. 2% of GDP. This parameer does no have any impac on he model behavior, bu i needs o be high enough o ensure han banks always have access o reasury bills o cover heir regulaory requiremens. Table 3: 1s order momens Daa Model Deposi raio Leverage raio Wholesale raio.12.1 Liquidiy Sress Raio Table 4: Sandard deviaionsin %) Daa Model Oupu Consumpion Invesmen Deposi raio Wholesale raio Leverage raio Liquidiy Sress Raio Toal bank asses Toal bank equiy

24 Table 5: Calibraion Parameer Value Descripion Sandard RBC parameers β.99 discoun facor α.36 share of capial in producion δ.25 depreciaion rae χ.5 adjusmen cos on invesmen χ d.8 adjusmen cos on deposis ϕ 1 inverse elasiciy of labor supply γ 1 inerermporal elasiciy of subsiuion η 1 disuiliy of labor z.58 seady sae TFP ρ z.9297 serial correlaion TFP shock σ z.67 sandard deviaion TFP shock Financial parameers σ.6988 seady sae idiosyncraic volailiy υ variance subsandard echnology ϑ.1 shif in mean of subsandard echnology τ.5846 share of asse ransfer ino new banks θ.75 survival probabiliy of banks ρ σ.9457 serial correlaion risk shock σ σ.465 sandard deviaion risk shock ι.5 deposi insurance fee parameer ρ κ.3591 serial correlaion capial qualiy shock σ κ.81 sandard deviaion capial qualiy shock ς.134 scaling parameer for all shocks Regulaory parameers ξ.5 seady sae regulaory parameer on deposis ξ 1.5 seady sae regulaory parameer on wholesale funding χ y,.5 cyclical regulaory parameer on deposis χ y,.5 cyclical regulaory parameer on wholesale funding T Bsupp 2 Toal supply of reasury bills 6.2 Seady-sae analysis The seady-sae values of he key variables of he model are given in able 6. Regulaion requires he bank o hold safe asses ha cover 5% of heir deposis and wholesale funding. In he unregulaed model, banks do no hold safe asses and he liquidiy raio is always zero. Indeed, since he reurn on reasury bills is 24

25 lower han he reurn on loans, he banks have no incenive o hold reasury bills unless hey are forced o do i by regulaion. Thus, liquidiy regulaion is always binding and implies an increase of he liquidiy raio. Firs, noe ha defaul probabiliies and defaul hresholds ω and ω are independen from liquidiy regulaion. ω is pinned down by he ICC, which simplifies o equaion 36). Higher defaul probabiliy makes he moral hazard more severe; hence a higher value of ω makes he ICC more binding, whereas low ω makes i slack. There is a unique value of ω ha makes he ICC hold wih equaliy. ω is pinned down by deposi demand of households and deposi supply of banks, which simplify o 37. Given our calibraion ξ = ξ 1 ), he seady-sae probabiliy of defaul on deposis depends only on ι. I follows ha defaul probabiliies in he unregulaed and regulaed models are idenical. Eω) Ẽω) = π ω) π ω) 36) F ω) = ι 1 ξ 1 ξ 1. 37) The porfolio of asses held by he regulaed bank is safer, since i holds a fracion in safe asses. We have seen ha lower risk of bank asses implies a reducion in he moral hazard, which means ha he bank can leverage more. The key poin is ha holding safe asses does no crowd ou credi o firms, bu raher crowds i in. The reason is ha he bank is able o leverage up while keeping he same probabiliy of defaul because is porfolio of asses is safer. Regulaed banks borrow more, hey increase boh deposis and wholesale funding. However hey increase deposis by more han wholesale funding, so ha heir deposi raio increases wih regulaion. Since he regulaed bank is holding more safe asses, i is able o increase deposis while mainaining an idenical probabiliy of defaul on deposis. The regulaion reduces he seady-sae value of he liquidiy sress raio. Banks are more leveraged, bu a he same ime hey carry less liquidiy mismach on heir balance shee. An imporan aspec of he model is how banks choose beween deposis and wholesale funding. Deposis are cheaper han wholesale funding, because he ineres rae on wholesale funding henceforh WS rae) is higher han he deposi rae. However, a higher deposi raio means higher probabiliy of defaul on de- 25

26 posis, and hereby higher deposi insurance paymen for he bank. Moreover, he WS rae increases wih he deposi raio. The reason is ha a high deposi raio implies a lower liquidaion value for he shadow bank when here is defaul. Since he shadow bank recuperaes less afer defaul, i requires a higher rae of reurn when here is no defaul, hence a higher WS rae. Banks face a rade-off: hey would prefer o use deposis ha are less cosly, bu he cos of boh deposi and wholesale funding is increasing in he deposi raio. Banks hen choose an allocaion of deposis and wholesale funding ha minimizes heir cos of funding. Table 6: Seady sae Unregulaed Regulaed Consumpion Labor Capial Oupu R A Liquidiy raio..452 Deposi raio Toal deposis Leverage raio Wholesale raio Toal Wholesale funding WS rae Liquidiy Sress Raio ω Defaul probabiliy ω Defaul on deposi probabiliy.5.5 Welfare Liquidiy regulaion has real effecs oo. An increase in regulaions resuls in more loans by banks. Thus, firms can have more invesmen and more capial. More capial implies higher marginal produciviy of labor, so ha labor goes up as well. Oupu increases, as well as consumpion. Lower marginal produc of capial implies lower ineres rae on he loans. Welfare is higher in he regulaed model, by.614% in consumpion equivalen erms. 26

27 6.3 Dynamic analysis We now urn our aenion o he dynamic behavior of he model. We consider hree differen shocks: risk shock, oal facor produciviy TFP) shock and capial qualiy shock. In each case, we compare he behavior of he unregulaed model, wih he models wih fla and cyclical liquidiy requiremen Response o a risk shock A risk shock is an increase in he cross-secional volailiy of he idiosyncraic capial qualiy shock. Since he disribuion is known one period in advance, he risk shock acs as a news shock: a ime he agens learn ha a +1 heir asses will become more risky. The impulse responses are in figure 4. Banks learn ha nex period heir asses are going o be more volaile. An increase in asse riskiness makes he moral hazard problem faced by he banks more severe. The shadow bank recognizes ha he bank has addiional incenives o inves subopimally and responds by cuing is lending o he bank. The oal amoun of wholesale funding B falls by abou 15% on impac, and banks are forced o deleverage. The defaul hreshold ω remains below seady sae from + 1 onwards. Since banks los a sizable porion of heir wholesale funding, hey end up wih a higher deposi raio. The probabiliy of defaul on deposi remains persisenly higher and so are deposi insurance paymens. This is an exra cos for he bank. Moreover, a higher deposi raio means a lower recuperaion value by he shadow bank when here is defaul. This implies ha he shadow bank imposes a higher WS rae. Thus, he banks have o pay more on boh heir deposis and heir wholesale funding. The LSR falls afer he shock, driven by he sharp fall in wholesale funding. Deleveraging implies ha banks need o curail credi o firms; hence, he risk shock has real effecs. Firms ge fewer loans from banks, hey need o cu invesmen and he price of capial falls. Marginal produciviy of labor also falls since capial is lower), so ha labor is lower. Oupu herefore falls as well. The fall in Q has an immediae impac on he reurn on loans a ime. R A falls on impac, which in urn affecs he curren values of ω and ω. Following a risk shock, invesmen falls more han oupu, so ha consumpion acually goes up. Following a risk shock, he model predics ha he wholesale funding raio is 27

28 Figure 4: The effec of liquidiy regulaion, risk shock 2 C I L K Y Q 5-1 R A WS rae ω bar defaul ω barbar defaul deposi deposi raio WS raio leverage LSR B D deposi insur unreg fla cyc pro-cyclical, he deposi raio is couner-cyclical and he liquidiy sress raio is pro-cyclical. This is consisen wih he sylized facs in secion 4. We now urn o he analysis of he dynamic implicaions of liquidiy regulaion. The LCR-ype fla liquidiy regulaion implies ha he banks need o keep a leas 5% of liquid asses agains deposis and wholesale funding a all ime. I is ineresing o noe ha while a fla liquidiy regulaion yields a seady sae wih higher consumpion and oupu, i acually amplifies flucuaions afer he shock. The shadow bank cus B more, which means a more severe deleveraging for he bank. The fall in oupu, invesmen and capial is also higher on impac, and akes longer o reurn o seady sae. Fla regulaion increases he persisency 28

29 in he model. The reason is primarily ha he regulaed and unregulaed models sar from differen seady saes. The effec of he risk shock is amplified due o he higher leverage in he regulaed model. Under he counercyclical liquidiy regulaion banks are required o hold more liquid asses when oupu is below seady sae. I may seem counerinuiive a firs o require bank o hold on o more safe asses during a recession, bu i has a sabilizing effec on he economy. Afer a risk shock, banks are forced o become safer, which relaxes heir moral hazard. As a resul, he shadow bank is less worried abou banks invesing subopimally, and he wholesale run is reduced. In oher words, he counercyclical liquidiy regulaion makes wholesale funding more sable over he cycle by reducing moral hazard exacly a he ime when i is mos acue. Noe ha defaul probabiliies a + 1 and onwards are he same for all models and are no affeced by regulaion. Wholesale funding raio falls less, so he deposi raio increases less, which in urn reduces he deposi insurance fee. Overall, he banks pay lower WS and deposi raes. The LSR is sill procyclical bu less so. On he asse side, banks do no need o cu loans as much. Since banks do no curail credi as much, he real variables are also less negaively affeced Response o a TFP shock The impulse response funcions following a negaive TFP shock are repored in figure 5. A TFP shock reduces he marginal produciviy of capial, and herefore he reurn on capial. On impac, banks ge a lower reurn on heir asses R A goes down), which drives up he defaul probabiliy a ime. Unlike he risk shock, a TFP shock does no affec he disribuion of ω and he ICC. Thus, he defaul hreshold ω +1 reurns o is seady-sae value from + 1 onwards. However, he TFP shock affecs he reurn on asses and hereby he profiabiliy of he banks. The shadow bank recognizes ha he banks are less profiable and responds by reducing wholesale lending by abou 5% on impac. The TFP shock and he wholesale run force banks o curail credi and keep a consan probabiliy of defaul from +1 onwards. The deposi raio increases; he probabiliy of defaul on deposi and insurance paymen also remain persisenly above seady sae. The recuperaion value for he shadow bank in case of bank defaul is lower, which leads o a higher WS rae. 29